Question

The ratio of two numbers $\frac{2}{3}$, If 2 is substracted from the first and 8 from the second, the ratio becomes the reciprocal of the original ratio. Find the numbers.

• A. 3 and 9
• B. 4 and 14
• C. 8 and 12
• D. 3 and 19

Answer 1

The correct option is C 8 and 12

Let the ratio be $\frac{x}{y}=\frac{2}{3}$ => $3x=2y$
$3x-2y=0$ --- (1)
Given,If 2 is subtracted from $x$ and 8 from $y$, then ratio $=\frac{x-2}{y-8}=\frac{3}{2}$
$=>2x-4=3y-24$
$=>2x-3y=-20$ --- (2)
Multiplying equation $\left(1\right)$ with $2$ we get, $6x-4y=0$ ----- equation $\left(3\right)$

Multiplying equation $\left(2\right)$ with $3$ we get, $6x-9y=-60$ ----- equation $\left(4\right)$

Subtracting equations $\left(4\right)$ from $\left(3\right)$, we get $5y=60$$=>y=12$

Substituting $y=12$ in the equation $\left(1\right)$, we get $3x-2\left(12\right)=0=>x=8$
Hence, the numbers are $8$ and $12$